Properties

Label 252.4.a.e
Level $252$
Weight $4$
Character orbit 252.a
Self dual yes
Analytic conductor $14.868$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8684813214\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -7 q^{7} +O(q^{10})\) \( q + \beta q^{5} -7 q^{7} -13 \beta q^{11} -30 q^{13} + 23 \beta q^{17} -100 q^{19} + 17 \beta q^{23} -97 q^{25} -44 \beta q^{29} -180 q^{31} -7 \beta q^{35} -118 q^{37} -3 \beta q^{41} -412 q^{43} + 54 \beta q^{47} + 49 q^{49} + 54 \beta q^{53} -364 q^{55} -158 \beta q^{59} -378 q^{61} -30 \beta q^{65} + 244 q^{67} + 83 \beta q^{71} + 670 q^{73} + 91 \beta q^{77} + 216 q^{79} + 152 \beta q^{83} + 644 q^{85} -183 \beta q^{89} + 210 q^{91} -100 \beta q^{95} + 574 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{7} + O(q^{10}) \) \( 2q - 14q^{7} - 60q^{13} - 200q^{19} - 194q^{25} - 360q^{31} - 236q^{37} - 824q^{43} + 98q^{49} - 728q^{55} - 756q^{61} + 488q^{67} + 1340q^{73} + 432q^{79} + 1288q^{85} + 420q^{91} + 1148q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.64575
2.64575
0 0 0 −5.29150 0 −7.00000 0 0 0
1.2 0 0 0 5.29150 0 −7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.a.e 2
3.b odd 2 1 inner 252.4.a.e 2
4.b odd 2 1 1008.4.a.bd 2
7.b odd 2 1 1764.4.a.t 2
7.c even 3 2 1764.4.k.t 4
7.d odd 6 2 1764.4.k.w 4
12.b even 2 1 1008.4.a.bd 2
21.c even 2 1 1764.4.a.t 2
21.g even 6 2 1764.4.k.w 4
21.h odd 6 2 1764.4.k.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.a.e 2 1.a even 1 1 trivial
252.4.a.e 2 3.b odd 2 1 inner
1008.4.a.bd 2 4.b odd 2 1
1008.4.a.bd 2 12.b even 2 1
1764.4.a.t 2 7.b odd 2 1
1764.4.a.t 2 21.c even 2 1
1764.4.k.t 4 7.c even 3 2
1764.4.k.t 4 21.h odd 6 2
1764.4.k.w 4 7.d odd 6 2
1764.4.k.w 4 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(252))\):

\( T_{5}^{2} - 28 \)
\( T_{11}^{2} - 4732 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -28 + T^{2} \)
$7$ \( ( 7 + T )^{2} \)
$11$ \( -4732 + T^{2} \)
$13$ \( ( 30 + T )^{2} \)
$17$ \( -14812 + T^{2} \)
$19$ \( ( 100 + T )^{2} \)
$23$ \( -8092 + T^{2} \)
$29$ \( -54208 + T^{2} \)
$31$ \( ( 180 + T )^{2} \)
$37$ \( ( 118 + T )^{2} \)
$41$ \( -252 + T^{2} \)
$43$ \( ( 412 + T )^{2} \)
$47$ \( -81648 + T^{2} \)
$53$ \( -81648 + T^{2} \)
$59$ \( -698992 + T^{2} \)
$61$ \( ( 378 + T )^{2} \)
$67$ \( ( -244 + T )^{2} \)
$71$ \( -192892 + T^{2} \)
$73$ \( ( -670 + T )^{2} \)
$79$ \( ( -216 + T )^{2} \)
$83$ \( -646912 + T^{2} \)
$89$ \( -937692 + T^{2} \)
$97$ \( ( -574 + T )^{2} \)
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